pith. sign in

arxiv: 2607.02203 · v1 · pith:AX336QV6new · submitted 2026-07-02 · 💻 cs.LG · physics.flu-dyn

Self-explainable Operator Learning for Discovering Spatial Patterns in Functional Data

Mojgan Alishiri , Amirhossein Arzani This is my paper

Pith reviewed 2026-07-03 16:45 UTC · model grok-4.3

classification 💻 cs.LG physics.flu-dyn
keywords self-explainable operator learningfunctional dataspatial interpretabilityintegral equationsneural operatorsfluid flowmodel transparencyfunction-to-function mapping
0
0 comments X

The pith

Reformulating neural operators as linear combinations of integral equations embeds direct spatial interpretability into the model itself.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Neural operators model complex physical systems in functional spaces yet remain opaque because their internal mappings are not readable. The paper shows that any such operator can be rewritten exactly as a sum of generalized functional linear models expressed through integral equations. The additive structure of those integrals then permits splitting the input domain into subdomains and computing each subdomain's separate contribution to the output. This decomposition directly connects particular input regions to the output patterns they produce, revealing which spatial features matter most. Demonstrations on blood-flow and unsteady-aerodynamics data confirm that the resulting explanations align with physical intuition while remaining intrinsic to the learned operator.

Core claim

The operator is reformulated as a linear combination of generalized functional linear models expressed through integral equations. Exploiting the additive decomposability of these integral equations, the input domain is divided into subdomains and localized integrals are computed to evaluate the contribution of each region to the final prediction. This decomposition enables direct interpretability where the model explains both inputs and outputs by linking specific input regions to corresponding output patterns, thereby revealing which spatial features drive predictions.

What carries the argument

Additive decomposition of the integral equations into localized subdomain contributions.

If this is right

  • Each input subdomain's localized integral directly quantifies its contribution to the output, producing an intrinsic explanation.
  • In the fluid examples the operator consistently assigns highest weight to regions containing strong feature gradients.
  • Explainability is obtained without any external post-hoc method and remains valid for both function-to-scalar and function-to-function tasks.
  • The same decomposition works for any operator that can be expressed in the required integral form while keeping predictive fidelity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be tested on operator-learning tasks outside fluids, such as heat transfer or structural mechanics, to check whether the gradient emphasis generalizes.
  • Direct comparison of the localized integrals against known analytical Green's functions in linear problems would provide an independent check on the decomposition.
  • If the method scales to high-dimensional inputs it might serve as a built-in feature-selection step before training larger operators.

Load-bearing premise

The neural operator can be rewritten as the stated linear combination of integral equations while preserving its original predictive accuracy and without introducing new fitting artifacts.

What would settle it

Apply both the original neural operator and the reformulated integral version to identical test inputs and observe whether the output predictions differ or whether the regions flagged as influential contradict established physical understanding of the flow problem.

Figures

Figures reproduced from arXiv: 2607.02203 by Amirhossein Arzani, Mojgan Alishiri.

Figure 1
Figure 1. Figure 1: XAI model training process. (a) Blood flow inside a brain aneurysm is chosen for func [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Post-processing procedure for the proposed XAI framework. (a) The input function domain for a representative sample from the blood flow function-to-function mapping task is parti￾tioned into four equal subdomains, Ω1–Ω4. Using the trained integral operator, the XAI model independently evaluates each subdomain to produce localized predictions (not mean-centered). Summing these four predictions recovers the … view at source ↗
Figure 3
Figure 3. Figure 3: Results of the function-to-scalar mapping task for predicting maximum WSS in the blood flow test case. (a) Velocity magnitude fields at selected test samples (S2–S11) spanning the pulsatile flow cycle. (b) Representative input velocity magnitude field (Sample S1) together with the four-zone domain decomposition used for the zonal contribution analysis. (c) Ground truth (black) and XAI model prediction (red… view at source ↗
Figure 4
Figure 4. Figure 4: Results of the function-to-scalar mapping task for (a) spatially averaged WSS (Mean WSS) and (b) spatial average of divergence of normalized WSS (Mean DNWSS) in the blood flow test case. Each panel shows the mean-centered zonal contributions from the four input subregions alongside the mean-centered ground truth (black) and XAI model prediction (red). Pearson correlation coefficients (PC) are indicated for… view at source ↗
Figure 5
Figure 5. Figure 5: Results of the function-to-function mapping task for the blood flow test case. (a) Maxi￾mum WSS versus time step for the test dataset, with six representative samples (S2–S6) marked. (b) Velocity magnitude fields at the corresponding samples, with the four-zone input domain decom￾position indicated. (c) Mean-centered zonal contributions from each input subregion at the selected samples, alongside the mean-… view at source ↗
Figure 6
Figure 6. Figure 6: Combinations of zonal contributions for the function-to-function mapping task at Sample 1 (marked in Fig. 5a). The first two rows show all possible two-zone and three-zone combinations of mean-centered zonal contributions. The bottom row shows the ground truth (not mean-centered and mean-centered) and the XAI model prediction (not mean-centered and mean-centered) for reference. a larger change indicates gr… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of post-hoc explainability methods for the function-to-function mapping task in the blood flow test case. (a) Input velocity fields, ground truth WSS outputs, and U-Net pre￾dictions for six representative test samples (S1–S6). The four-zone input domain decomposition is indicated on sample S1. (b) Zonal importance scores from Kernel SHAP (top row) and occlusion sensitivity (bottom row). Each 2 ×… view at source ↗
Figure 8
Figure 8. Figure 8: Results of the function-to-scalar mapping task for the airfoil test case at nondimensional pitching frequency fp = 0.05. (a) Time variation of the drag coefficient with selected representative samples highlighted along the cycle. (b) Velocity magnitude fields corresponding to the selected time steps, together with the four-zone decomposition of the input domain. (c) Mean-centered zonal contributions for th… view at source ↗
Figure 9
Figure 9. Figure 9: Results of the function-to-scalar mapping task for the airfoil test case at nondimensional pitching frequency fp = 0.2. (a) Time variation of the drag coefficient with selected representative samples highlighted along the cycle. (b) Velocity magnitude fields corresponding to the selected time steps, together with the four-zone decomposition of the input domain. (c) Mean-centered zonal contributions for the… view at source ↗
Figure 10
Figure 10. Figure 10: Sensitivity of the zonal decomposition to the number of input subregions k for the function-to-function mapping task in the blood flow test case. (a) Decomposition using k = 2 with a horizontal partition of the input domain. (b) Decomposition using k = 2 with a vertical partition. (c) Decomposition using k = 9 using a 3 × 3 grid partition. For each configuration, representative mean-centered zonal contrib… view at source ↗
Figure 11
Figure 11. Figure 11: Grad-CAM heatmaps for the function-to-function mapping task in the blood flow test case, computed at four encoder stages of the U-Net. The top row shows the input velocity fields for six representative test samples (S1–S6). Subsequent rows display the Grad-CAM saliency maps at Encoder Stages 1–4, with spatial resolutions of 18 × 18, 9 × 9, 4 × 4, and 2 × 2, respectively. All heatmaps are upsampled to the … view at source ↗
Figure 12
Figure 12. Figure 12: Sensitivity analysis for the Max WSS function-to-scalar mapping in the blood flow test case. (a) Effect of varying βmax and λ with βmin fixed at 0.1: Mean AE (left), Max AE (center), and distribution of zonal Pearson correlations across all configurations (right). The star marks the baseline setting used in the main text. (b) Effect of varying βmin and λ with βmax fixed at 2.5. N/A cells indicate unstable… view at source ↗
Figure 13
Figure 13. Figure 13: Combinations of zonal contributions for the three function-to-scalar mapping tasks in the blood flow test case: (a) maximum WSS, (b) spatially averaged WSS (Mean WSS), and (c) spatial average of the divergence of normalized WSS (Mean DNWSS). The first column shows the best two-zone combination (blue), and the second column shows the best three-zone combination (blue), alongside the mean-centered ground tr… view at source ↗
Figure 14
Figure 14. Figure 14: Combinations of zonal contributions for the airfoil function-to-scalar mapping tasks. (a) Drag coefficient prediction at nondimensional pitching frequency fp = 0.05. (b) Lift coefficient prediction at fp = 0.05. (c) Drag coefficient prediction at fp = 0.2. (d) Lift coefficient prediction at fp = 0.2. In each subfigure, the left panel shows the best-performing two-zone combination and the right panel shows… view at source ↗
Figure 15
Figure 15. Figure 15: Time-varying analysis of zonal contributions for the function-to-function mapping task in the blood flow test case. Spatially averaged WSS values are computed from the output fields at each test time step. The first two panels show the mean-centered ground truth and XAI model prediction. The next four panels show the mean-centered contribution of each zone, with Pearson correlation coefficients (PC) indic… view at source ↗
Figure 16
Figure 16. Figure 16: Results of the additional function-to-function mapping experiment using the synthetic semi-linear elliptic PDE dataset. The first column shows representative input functions for selected test samples. The second and third columns show the corresponding ground truth outputs and XAI model predictions, respectively. Columns 4–7 present the mean-centered zonal contributions obtained from the four input subreg… view at source ↗
read the original abstract

Operator learning has emerged as a powerful tool for modeling complex physical systems in functional spaces. However, their neural network-based architectures make them opaque models, obscuring the reasoning behind their predictions. In this work, we introduce a self-explainable operator learning framework that overcomes this challenge by reformulating operator learning as a linear combination of generalized functional linear models expressed through integral equations. Exploiting the additive decomposability of these integral equations, we divide the input domain into subdomains and compute localized integrals to evaluate the contribution of each region to the final prediction. This decomposition enables direct interpretability where the model explains both inputs and outputs by linking specific input regions to corresponding output patterns, thereby revealing which spatial features drive predictions. We demonstrate the framework on function-to-scalar and function-to-function mappings in fluid flow problems involving blood flow and unsteady aerodynamics. The results show that the operator most often prioritizes regions with strong feature gradients, providing physically meaningful insight into the model's decision-making process. Comparisons with established post-hoc explainability methods demonstrate qualitative agreement while highlighting the key advantage of the proposed approach: explainability is embedded directly within the operator structure itself and does not require an external tool. Therefore, our framework provides a mathematically transparent and physically interpretable approach to uncover relationships within data, fostering trust in machine learning for scientific applications by enabling more informed data-driven analysis of physical systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript proposes a self-explainable operator learning framework that reformulates neural operators as linear combinations of generalized functional linear models expressed through integral equations. By exploiting additive decomposability, the input domain is partitioned into subdomains whose localized integrals quantify each region's contribution to the output prediction. The approach is demonstrated on function-to-scalar and function-to-function mappings for blood-flow and unsteady-aerodynamics problems, with the claim that attributions align with strong gradients, agree qualitatively with post-hoc explainers, and are obtained intrinsically without external tools.

Significance. If the reformulation is faithful to the original nonlinear operator class and the decomposition introduces no measurable degradation in predictive accuracy, the framework would supply a mathematically grounded route to intrinsic interpretability for scientific operator learning, reducing dependence on post-hoc attribution methods.

major comments (3)
  1. [Abstract] Abstract: the central claim that the operator can be rewritten as a linear combination of generalized functional linear models while preserving original predictive accuracy is unsupported by any reported error metrics, ablation of the decomposition step, or side-by-side comparison of the original versus reformulated model on the blood-flow or aerodynamics tasks.
  2. [Abstract] Abstract: no equations, kernel definitions, or fitting procedure are supplied for the subdomain integrals or linear coefficients, leaving open whether the decomposition is exact for the target operator class or an approximation whose error could be misread as physically meaningful spatial attributions.
  3. [Abstract] Abstract: the reported demonstrations show gradient-aligned attributions but supply neither quantitative fidelity measures (e.g., correlation with known physical features, perturbation-based explanation tests) nor verification that the attributions remain stable under changes to the subdomain partition.
minor comments (1)
  1. The abstract would be strengthened by naming the concrete neural-operator architectures (FNO, DeepONet, etc.) to which the reformulation is applied.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address each major comment below and indicate where revisions will be made to strengthen the presentation of the self-explainable operator learning framework.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the operator can be rewritten as a linear combination of generalized functional linear models while preserving original predictive accuracy is unsupported by any reported error metrics, ablation of the decomposition step, or side-by-side comparison of the original versus reformulated model on the blood-flow or aerodynamics tasks.

    Authors: The reformulation is constructed to be exact for the target operator class by exploiting the additive decomposability of the integral equations, implying preservation of predictive accuracy by design. We acknowledge, however, that explicit empirical verification strengthens the claim. In the revised manuscript we will add a direct comparison of predictive errors (relative L2 norms) between the original neural operator and the reformulated self-explainable version on both the blood-flow and aerodynamics tasks, together with an ablation isolating the decomposition step. revision: yes

  2. Referee: [Abstract] Abstract: no equations, kernel definitions, or fitting procedure are supplied for the subdomain integrals or linear coefficients, leaving open whether the decomposition is exact for the target operator class or an approximation whose error could be misread as physically meaningful spatial attributions.

    Authors: Abstracts conventionally omit detailed equations to preserve readability. The full manuscript supplies the integral-equation reformulation, kernel definitions, and the procedure for obtaining the linear coefficients in Sections 2 and 3; the decomposition is exact by the additive property of the generalized functional linear models. We will revise the abstract to include a brief pointer to these sections. revision: partial

  3. Referee: [Abstract] Abstract: the reported demonstrations show gradient-aligned attributions but supply neither quantitative fidelity measures (e.g., correlation with known physical features, perturbation-based explanation tests) nor verification that the attributions remain stable under changes to the subdomain partition.

    Authors: We agree that quantitative fidelity and stability analyses would provide stronger support. The current demonstrations already show qualitative agreement with gradients and post-hoc explainers. In the revision we will add Pearson correlations between attribution maps and known physical features (gradient magnitudes), perturbation-based tests, and results demonstrating attribution stability across multiple subdomain partitions for both application domains. revision: yes

Circularity Check

0 steps flagged

No significant circularity; reformulation is an architectural choice, not a reduction to fitted inputs.

full rationale

The paper proposes reformulating neural operators as linear combinations of generalized functional linear models via integral equations to embed subdomain-based interpretability. This is presented as a direct modeling decision that exploits additive decomposability, with explanations arising by construction from the chosen structure rather than from any post-hoc fitting or self-referential derivation. No load-bearing self-citation, uniqueness theorem, or fitted parameter renamed as a prediction is evident. The central claim of embedded explainability without external tools follows from the framework definition itself and remains independent of the training data or prior results. Demonstrations are qualitative and do not reduce any claimed spatial attribution to a tautological fit.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review limits visibility into parameters and assumptions; the central reformulation itself functions as the primary domain assumption.

axioms (1)
  • domain assumption Operator learning models can be reformulated as linear combinations of generalized functional linear models expressed through integral equations.
    This reformulation is the foundational step stated in the abstract that enables the subsequent decomposition.

pith-pipeline@v0.9.1-grok · 5779 in / 1251 out tokens · 33880 ms · 2026-07-03T16:45:30.740691+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

85 extracted references · 14 canonical work pages · 7 internal anchors

  1. [1]

    Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead

    C. Rudin. “Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead”. In:Nature machine intelligence1.5 (2019), pp. 206–215

    2019

  2. [2]

    Interpretability of machine-learning models in physical sciences

    L. M. Ghiringhelli. “Interpretability of machine-learning models in physical sciences”. In: arXiv preprint arXiv:2104.10443(2021)

    work page arXiv 2021

  3. [3]

    Scientific inference with in- terpretable machine learning: Analyzing models to learn about real-world phenomena

    T. Freiesleben, G. K¨ onig, C. Molnar, and A. Tejero-Cantero. “Scientific inference with in- terpretable machine learning: Analyzing models to learn about real-world phenomena”. In: Minds and Machines34.3 (2024), p. 32

    2024

  4. [4]

    Interpretable models for extrapolation in scientific machine learning

    E. S. Muckley, J. E. Saal, B. Meredig, C. S. Roper, and J. H. Martin. “Interpretable models for extrapolation in scientific machine learning”. In:Digital Discovery2.5 (2023), pp. 1425– 1435

    2023

  5. [5]

    Learning operators with coupled attention

    G. Kissas, J. H. Seidman, L. F. Guilhoto, V. M. Preciado, G. J. Pappas, and P. Perdikaris. “Learning operators with coupled attention”. In:Journal of Machine Learning Research 23.215 (2022), pp. 1–63

    2022

  6. [6]

    Neural operator: Learning maps between function spaces with applications to PDEs

    N. Kovachki, Z. Li, B. Liu, K. Azizzadenesheli, K. Bhattacharya, A. Stuart, and A. Anandku- mar. “Neural operator: Learning maps between function spaces with applications to PDEs”. In:Journal of Machine Learning Research24.89 (2023), pp. 1–97

    2023

  7. [7]

    An explainable operator approximation framework un- der the guideline of Green’s function

    J. Gu, L. Wen, Y. Chen, and S. Chen. “An explainable operator approximation framework un- der the guideline of Green’s function”. In:Journal of Computational Physics(2025), p. 114244

    2025

  8. [8]

    How to understand masked autoencoders

    S. Cao, P. Xu, and D. A. Clifton. “How to understand masked autoencoders”. In:arXiv preprint arXiv:2202.03670(2022)

    work page arXiv 2022

  9. [9]

    Scalable transformer for PDE surrogate modeling

    Z. Li, D. Shu, and A. Barati Farimani. “Scalable transformer for PDE surrogate modeling”. In:Advances in Neural Information Processing Systems36 (2023), pp. 28010–28039

    2023

  10. [10]

    Interpreting and generalizing deep learning in physics-based problems with functional linear models

    A. Arzani, L. Yuan, P. Newell, and B. Wang. “Interpreting and generalizing deep learning in physics-based problems with functional linear models”. In:Engineering with Computers41.1 (2025), pp. 135–157

    2025

  11. [11]

    Horv´ ath and P

    L. Horv´ ath and P. Kokoszka.Inference for functional data with applications. Vol. 200. Springer Science & Business Media, 2012

    2012

  12. [12]

    Fourier Neural Operator for Parametric Partial Differential Equations

    Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anandku- mar. “Fourier neural operator for parametric partial differential equations”. In:arXiv preprint arXiv:2010.08895(2020)

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  13. [13]

    DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators

    L. Lu, P. Jin, and G. E. Karniadakis. “Deeponet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators”. In:arXiv preprint arXiv:1910.03193(2019)

    work page internal anchor Pith review Pith/arXiv arXiv 1910

  14. [14]

    The mythos of model interpretability: In machine learning, the concept of interpretability is both important and slippery

    Z. C. Lipton. “The mythos of model interpretability: In machine learning, the concept of interpretability is both important and slippery.” In:Queue16.3 (2018), pp. 31–57

    2018

  15. [15]

    Explainable Artificial Intelligence: Understanding, Visualizing and Interpreting Deep Learning Models

    W. Samek, T. Wiegand, and K.-R. M¨ uller. “Explainable artificial intelligence: Understand- ing, visualizing and interpreting deep learning models”. In:arXiv preprint arXiv:1708.08296 (2017)

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  16. [16]

    Kernel methods in machine learning

    T. Hofmann, B. Sch¨ olkopf, and A. J. Smola. “Kernel methods in machine learning”. In:The Annals of Statistics36.3 (2008), pp. 1171–1220. p. 45

    2008

  17. [17]

    Explainable machine learning for scientific insights and discoveries

    R. Roscher, B. Bohn, M. F. Duarte, and J. Garcke. “Explainable machine learning for scientific insights and discoveries”. In:Ieee Access8 (2020), pp. 42200–42216

    2020

  18. [18]

    Interpretable machine learning: definitions, methods, and applications

    W. J. Murdoch, C. Singh, K. Kumbier, R. Abbasi-Asl, and B. Yu. “Interpretable machine learning: definitions, methods, and applications”. In:arXiv preprint arXiv:1901.04592(2019)

    work page arXiv 1901

  19. [19]

    Interpretable and explainable machine learning: A methods- centric overview with concrete examples

    R. Marcinkeviˇ cs and J. E. Vogt. “Interpretable and explainable machine learning: A methods- centric overview with concrete examples”. In:Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery13.3 (2023), e1493

    2023

  20. [20]

    Informed machine learning–a taxonomy and sur- vey of integrating prior knowledge into learning systems

    L. Von Rueden, S. Mayer, K. Beckh, B. Georgiev, S. Giesselbach, R. Heese, B. Kirsch, J. Pfrommer, A. Pick, R. Ramamurthy, et al. “Informed machine learning–a taxonomy and sur- vey of integrating prior knowledge into learning systems”. In:IEEE Transactions on Knowl- edge and Data Engineering35.1 (2021), pp. 614–633

    2021

  21. [21]

    Automated, predictive, and interpretable in- ference of Caenorhabditis elegans escape dynamics

    B. C. Daniels, W. S. Ryu, and I. Nemenman. “Automated, predictive, and interpretable in- ference of Caenorhabditis elegans escape dynamics”. In:Proceedings of the National Academy of Sciences116.15 (2019), pp. 7226–7231

    2019

  22. [22]

    Learning integral operators via neural integral equations

    E. Zappala, A. H. d. O. Fonseca, J. O. Caro, A. H. Moberly, M. J. Higley, J. Cardin, and D. v. Dijk. “Learning integral operators via neural integral equations”. In:Nature Machine Intelligence6.9 (2024), pp. 1046–1062

    2024

  23. [23]

    Interpretable learning of effective dynamics for multiscale systems

    E. Menier, S. Kaltenbach, M. Yagoubi, M. Schoenauer, and P. Koumoutsakos. “Interpretable learning of effective dynamics for multiscale systems”. In:Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences481.2305 (2025)

    2025

  24. [24]

    Design of distributed rule-based models in the presence of large data

    E Hanyu, Y. Cui, W. Pedrycz, and Z. Li. “Design of distributed rule-based models in the presence of large data”. In:IEEE Transactions on Fuzzy Systems31.7 (2022), pp. 2479–2486

    2022

  25. [25]

    Supersparse linear integer models for optimized medical scoring systems

    B. Ustun and C. Rudin. “Supersparse linear integer models for optimized medical scoring systems”. In:Machine Learning102.3 (2016), pp. 349–391

    2016

  26. [26]

    Generalized additive models

    T. Hastie and R. Tibshirani. “Generalized additive models”. In:Statistical science1.3 (1986), pp. 297–310

    1986

  27. [27]

    Sparse additive models

    P. Ravikumar, J. Lafferty, H. Liu, and L. Wasserman. “Sparse additive models”. In:Journal of the Royal Statistical Society Series B: Statistical Methodology71.5 (2009), pp. 1009–1030

    2009

  28. [28]

    Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl

    M. Cranmer. “Interpretable machine learning for science with PySR and SymbolicRegression. jl”. In:arXiv preprint arXiv:2305.01582(2023)

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  29. [29]

    Explainable AI: A review of machine learning interpretability methods

    P. Linardatos, V. Papastefanopoulos, and S. Kotsiantis. “Explainable AI: A review of machine learning interpretability methods”. In:Entropy23.1 (2020), p. 18

    2020

  30. [30]

    Feature importance ranking for deep learning

    M. Wojtas and K. Chen. “Feature importance ranking for deep learning”. In:Advances in neural information processing systems33 (2020), pp. 5105–5114

    2020

  31. [31]

    ” Why should I trust you?

    M. T. Ribeiro, S. Singh, and C. Guestrin. “” Why should I trust you?” Explaining the predic- tions of any classifier”. In:Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining. 2016, pp. 1135–1144

    2016

  32. [32]

    A unified approach to interpreting model predictions

    S. M. Lundberg and S.-I. Lee. “A unified approach to interpreting model predictions”. In: Advances in neural information processing systems30 (2017)

    2017

  33. [33]

    Axiomatic attribution for deep networks

    M. Sundararajan, A. Taly, and Q. Yan. “Axiomatic attribution for deep networks”. In:Inter- national conference on machine learning. PMLR. 2017, pp. 3319–3328. p. 46

    2017

  34. [34]

    Explainable AI (XAI): Core ideas, techniques, and solutions

    R. Dwivedi, D. Dave, H. Naik, S. Singhal, R. Omer, P. Patel, B. Qian, Z. Wen, T. Shah, G. Morgan, et al. “Explainable AI (XAI): Core ideas, techniques, and solutions”. In:ACM computing surveys55.9 (2023), pp. 1–33

    2023

  35. [35]

    Visualizing and understanding convolutional networks

    M. D. Zeiler and R. Fergus. “Visualizing and understanding convolutional networks”. In: European conference on computer vision. Springer. 2014, pp. 818–833

    2014

  36. [36]

    Grad-CAM: visual explanations from deep networks via gradient-based localization

    R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, and D. Batra. “Grad-CAM: visual explanations from deep networks via gradient-based localization”. In:International journal of computer vision128.2 (2020), pp. 336–359

    2020

  37. [37]

    Evaluation of post-hoc interpretability methods in time-series classification

    H. Turb´ e, M. Bjelogrlic, C. Lovis, and G. Mengaldo. “Evaluation of post-hoc interpretability methods in time-series classification”. In:Nature Machine Intelligence5.3 (2023), pp. 250– 260

    2023

  38. [38]

    The level of strength of an explanation: A quantitative evaluation technique for post-hoc XAI methods

    M. Bello, R. Amador, M.-M. Garc´ ıa, J. Del Ser, P. Mesejo, and ´O. Cord´ on. “The level of strength of an explanation: A quantitative evaluation technique for post-hoc XAI methods”. In:Pattern Recognition161 (2025), p. 111221

    2025

  39. [39]

    Identifying regions of importance in wall- bounded turbulence through explainable deep learning

    A. Cremades, S. Hoyas, R. Deshpande, P. Quintero, M. Lellep, W. J. Lee, J. P. Monty, N. Hutchins, M. Linkmann, I. Marusic, et al. “Identifying regions of importance in wall- bounded turbulence through explainable deep learning”. In:Nature Communications15.1 (2024), p. 3864

    2024

  40. [40]

    Explainable AI in aerospace for enhanced system performance

    S. Sutthithatip, S. Perinpanayagam, S. Aslam, and A. Wileman. “Explainable AI in aerospace for enhanced system performance”. In:2021 IEEE/AIAA 40th Digital Avionics Systems Con- ference (DASC). IEEE. 2021, pp. 1–7

    2021

  41. [41]

    Explainable AI in drug discovery: self-interpretable graph neural network for molecular property prediction using concept whitening

    M. Proietti, A. Ragno, B. L. Rosa, R. Ragno, and R. Capobianco. “Explainable AI in drug discovery: self-interpretable graph neural network for molecular property prediction using concept whitening”. In:Machine Learning113.4 (2024), pp. 2013–2044

    2024

  42. [42]

    Optimization of wearable biosensor data for stress classification using machine learning and explainable AI

    D. Sethia, S Indu, et al. “Optimization of wearable biosensor data for stress classification using machine learning and explainable AI”. In:IEEE Access(2024)

    2024

  43. [43]

    Scientific exploration and explainable artificial intelligence

    C. Zednik and H. Boelsen. “Scientific exploration and explainable artificial intelligence”. In: Minds and Machines32.1 (2022), pp. 219–239

    2022

  44. [44]

    From kepler to newton: Explainable AI for science discovery

    Z. Li, J. Ji, and Y. Zhang. “From kepler to newton: Explainable AI for science discovery”. In: arXiv preprint arXiv:2111.12210(2021)

    work page arXiv 2021

  45. [45]

    Explainable AI-driven IoMT fusion: Unravelling techniques, opportunities, and challenges with Explainable AI in healthcare

    N. A. Wani, R. Kumar, J. Bedi, I. Rida, et al. “Explainable AI-driven IoMT fusion: Unravelling techniques, opportunities, and challenges with Explainable AI in healthcare”. In:Information Fusion110 (2024), p. 102472

    2024

  46. [46]

    Explainable artificial intelligence for botnet detection in internet of things

    M. Saied and S. Guirguis. “Explainable artificial intelligence for botnet detection in internet of things”. In:Scientific Reports15.1 (2025), p. 7632

    2025

  47. [47]

    SFX-GAN: sustainable and explainable multi-modal spectral fusion for image dehazing in complex systems

    P. Dwivedia, V. Agarwal, N. A. Wani, M. Kajal, and A. Bennour. “SFX-GAN: sustainable and explainable multi-modal spectral fusion for image dehazing in complex systems”. In:Complex & Intelligent Systems(2026)

    2026

  48. [48]

    How can I choose an explainer? An application-grounded evaluation of post-hoc explanations

    S. Jesus, C. Bel´ em, V. Balayan, J. Bento, P. Saleiro, P. Bizarro, and J. Gama. “How can I choose an explainer? An application-grounded evaluation of post-hoc explanations”. In: Proceedings of the 2021 ACM conference on fairness, accountability, and transparency. 2021, pp. 805–815. p. 47

    2021

  49. [49]

    On the tractability of SHAP explanations

    G. Van den Broeck, A. Lykov, M. Schleich, and D. Suciu. “On the tractability of SHAP explanations”. In:Journal of Artificial Intelligence Research74 (2022), pp. 851–886

    2022

  50. [50]

    Fooling LIME and SHAP: Adversar- ial attacks on post hoc explanation methods

    D. Slack, S. Hilgard, E. Jia, S. Singh, and H. Lakkaraju. “Fooling LIME and SHAP: Adversar- ial attacks on post hoc explanation methods”. In:Proceedings of the AAAI/ACM Conference on AI, Ethics, and Society. 2020, pp. 180–186

    2020

  51. [51]

    Kernel Neural Operators (KNOs) for Scalable, Memory-efficient, Geometrically-flexible Operator Learning

    M. Lowery, J. Turnage, Z. Morrow, J. D. Jakeman, A. Narayan, S. Zhe, and V. Shankar. “Kernel neural operators (KNOs) for scalable, memory-efficient, geometrically-flexible opera- tor learning”. In:arXiv preprint arXiv:2407.00809(2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  52. [52]

    Functional linear model

    H. Cardot, F. Ferraty, and P. Sarda. “Functional linear model”. In:Statistics & Probability Letters45.1 (1999), pp. 11–22

    1999

  53. [53]

    Functional data analysis

    J.-L. Wang, J.-M. Chiou, and H.-G. M¨ uller. “Functional data analysis”. In:Annual Review of Statistics and its application3.1 (2016), pp. 257–295

    2016

  54. [54]

    P. J. Phillips, C. A. Hahn, P. C. Fontana, A. Yates, K. Greene, D. A. Broniatowski, and M. A. Przybocki.Four principles of explainable artificial intelligence. Tech. rep. National Institute of Standards and Technology, 2021

    2021

  55. [55]

    Functional generalized additive models

    M. W. McLean, G. Hooker, A.-M. Staicu, F. Scheipl, and D. Ruppert. “Functional generalized additive models”. In:Journal of Computational and Graphical Statistics23.1 (2014), pp. 249– 269

    2014

  56. [56]

    Lift & learn: Physics-informed machine learning for large-scale nonlinear dynamical systems

    E. Qian, B. Kramer, B. Peherstorfer, and K. Willcox. “Lift & learn: Physics-informed machine learning for large-scale nonlinear dynamical systems”. In:Physica D: Nonlinear Phenomena 406 (2020), p. 132401

    2020

  57. [57]

    Exact and inexact lifting transformations of nonlinear dynamical systems: Transfer functions, equivalence, and complexity reduction

    I. V. Gosea. “Exact and inexact lifting transformations of nonlinear dynamical systems: Transfer functions, equivalence, and complexity reduction”. In:Applied Sciences12.5 (2022), p. 2333

    2022

  58. [58]

    ADAM-SINDy: An efficient optimiza- tion framework for parameterized nonlinear dynamical system identification

    S. Viknesh, Y. Tatari, C. Christenson, and A. Arzani. “ADAM-SINDy: An efficient optimiza- tion framework for parameterized nonlinear dynamical system identification”. In:Physical review research8.1 (2026), p. 013040

    2026

  59. [59]

    Analysis of fluid flows via spectral properties of the Koopman operator

    I. Mezi´ c. “Analysis of fluid flows via spectral properties of the Koopman operator”. In:Annual review of fluid mechanics45.1 (2013), pp. 357–378

    2013

  60. [60]

    A kernel-based method for data-driven Koopman spectral analysis

    M. O. Williams, C. W. Rowley, and I. G. Kevrekidis. “A kernel-based method for data-driven Koopman spectral analysis”. In:Journal of Computational Dynamics2.2 (2015), pp. 247–265. doi:10.3934/jcd.2015005

    work page doi:10.3934/jcd.2015005 2015

  61. [61]

    A review and comparison of bandwidth selection methods for kernel regression

    M. K¨ ohler, A. Schindler, and S. Sperlich. “A review and comparison of bandwidth selection methods for kernel regression”. In:International Statistical Review82.2 (2014), pp. 243–274

    2014

  62. [62]

    Bandwidth selection for kernel distribution function estimation

    N. Altman and C. Leger. “Bandwidth selection for kernel distribution function estimation”. In:Journal of Statistical Planning and Inference46.2 (1995), pp. 195–214

    1995

  63. [63]

    Molnar.Interpretable machine learning

    C. Molnar.Interpretable machine learning. Lulu. com, 2020

    2020

  64. [64]

    U-net: Convolutional networks for biomedical im- age segmentation

    O. Ronneberger, P. Fischer, and T. Brox. “U-net: Convolutional networks for biomedical im- age segmentation”. In:International Conference on Medical image computing and computer- assisted intervention. Springer. 2015, pp. 234–241

    2015

  65. [65]

    Comparing different nonlinear dimensionality reduction techniques for data-driven unsteady fluid flow modeling

    H. Csala, S. Dawson, and A. Arzani. “Comparing different nonlinear dimensionality reduction techniques for data-driven unsteady fluid flow modeling”. In:Physics of Fluids34.11 (2022). p. 48

    2022

  66. [66]

    The story of wall shear stress in coronary artery atherosclerosis: biochemical transport and mechanotransduction

    M. Mahmoudi, A. Farghadan, D. R. McConnell, A. J. Barker, J. J. Wentzel, M. J. Budoff, and A. Arzani. “The story of wall shear stress in coronary artery atherosclerosis: biochemical transport and mechanotransduction”. In:Journal of biomechanical engineering143.4 (2021), p. 041002

    2021

  67. [67]

    Wall shear stress topological skeleton independently predicts long-term restenosis after carotid bifurcation endarterectomy

    U. Morbiducci, V. Mazzi, M. Domanin, G. De Nisco, C. Vergara, D. A. Steinman, and D. Gallo. “Wall shear stress topological skeleton independently predicts long-term restenosis after carotid bifurcation endarterectomy”. In:Annals of biomedical engineering48.12 (2020), pp. 2936–2949

    2020

  68. [68]

    Drag prediction of rough-wall turbulent flow using data-driven regression

    Z. Shi, S. M. H. Khorasani, H. Shin, J. Yang, S. Lee, and S. Bagheri. “Drag prediction of rough-wall turbulent flow using data-driven regression”. In:Flow5 (2025), E5

    2025

  69. [69]

    A database for reduced-complexity modeling of fluid flows

    A. Towne, S. T. Dawson, G. A. Br` es, A. Lozano-Dur´ an, T. Saxton-Fox, A. Parthasarathy, A. R. Jones, H. Biler, C.-A. Yeh, H. D. Patel, et al. “A database for reduced-complexity modeling of fluid flows”. In:AIAA journal61.7 (2023), pp. 2867–2892

    2023

  70. [70]

    Neural operators with localized integral and differential kernels

    M. Liu-Schiaffini, J. Berner, B. Bonev, T. Kurth, K. Azizzadenesheli, and A. Anandkumar. “Neural operators with localized integral and differential kernels”. In:Proceedings of the 41st International Conference on Machine Learning (ICML). 2024, pp. 32576–32594

    2024

  71. [71]

    Discovering governing equations from data by sparse identification of nonlinear dynamical systems

    S. L. Brunton, J. L. Proctor, and J. N. Kutz. “Discovering governing equations from data by sparse identification of nonlinear dynamical systems”. In:Proceedings of the national academy of sciences113.15 (2016), pp. 3932–3937

    2016

  72. [72]

    Neural additive models: Interpretable machine learning with neural nets

    R. Agarwal, L. Melnick, N. Frosst, X. Zhang, B. Lengerich, R. Caruana, and G. E. Hinton. “Neural additive models: Interpretable machine learning with neural nets”. In:Advances in neural information processing systems34 (2021), pp. 4699–4711

    2021

  73. [73]

    Neural Operator: Graph Kernel Network for Partial Differential Equations

    Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anand- kumar. “Neural operator: Graph kernel network for partial differential equations”. In:arXiv preprint arXiv:2003.03485(2020)

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  74. [74]

    C. K. Williams and C. E. Rasmussen.Gaussian processes for machine learning. Vol. 2. 3. MIT press Cambridge, MA, 2006

    2006

  75. [75]

    Deep Convolutional Networks as shal- low Gaussian Processes

    A. Garriga-Alonso, C. E. Rasmussen, and L. Aitchison. “Deep Convolutional Networks as shal- low Gaussian Processes”. In:International Conference on Learning Representations (ICLR). 2019

    2019

  76. [76]

    Discriminative feature attributions: Bridging post hoc explainability and inherent interpretability

    U. Bhalla, S. Srinivas, and H. Lakkaraju. “Discriminative feature attributions: Bridging post hoc explainability and inherent interpretability”. In:Advances in Neural Information Process- ing Systems36 (2023), pp. 44105–44122

    2023

  77. [77]

    Sanity checks for saliency maps

    J. Adebayo, J. Gilmer, M. Muelly, I. Goodfellow, M. Hardt, and B. Kim. “Sanity checks for saliency maps”. In:Advances in neural information processing systems31 (2018)

    2018

  78. [78]

    The (un)reliability of saliency methods

    P.-J. Kindermans, S. Hooker, J. Adebayo, M. Alber, K. T. Sch¨ utt, S. D¨ ahne, D. Erhan, and B. Kim. “The (un)reliability of saliency methods”. In:Explainable AI: Interpreting, Explaining and Visualizing Deep Learning. Springer, 2019, pp. 267–280

    2019

  79. [79]

    RISE: Randomized input sampling for explanation of black-box models

    V. Petsiuk, A. Das, and K. Saenko. “RISE: Randomized input sampling for explanation of black-box models”. In:British Machine Vision Conference (BMVC). 2018

    2018

  80. [80]

    Interpretable explanations of black boxes by meaningful per- turbation

    R. C. Fong and A. Vedaldi. “Interpretable explanations of black boxes by meaningful per- turbation”. In:Proceedings of the IEEE international conference on computer vision. 2017, pp. 3429–3437. p. 49

    2017

Showing first 80 references.